View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by TED Ankara College IB Thesis MATHEMATICS EXTENDED ESSAY “Relationship Between Mathematics and Art” Candidate Name: Zeynep AYGAR Candidate Number: 1129-018 Session: MAY 2013 Supervisor: Mehmet Emin ÖZER TED Ankara College Foundation High School ZEYNEP AYGAR D1129018 ABSTRACT For most of the people, mathematics is a school course with symbols and rules, which they suffer understanding and find difficult throughout their educational life, and even they find it useless in their daily life. Mathematics and arts, in general, are considered and put apart from each other. Mathematics represents truth, while art represents beauty. Mathematics has theories and proofs, but art depends on personal thought. Even though mathematics is a combination of symbols and rules, it has strong effects on daily life and art. It is obvious that those people dealing with strict rules of positive sciences only but nothing else suffer from the lack of emotion. Art has no meaning for them. On the other hand, artists unaware of the rules controlling the whole universe are living in a utopic world. However, a man should be aware both of the real world and emotional world in order to become happy in his life. In order to achieve that he needs two things: mathematics and art. Mathematics is the reflection of the physical world in human mind. Art is a mirror of human spirit. While mathematics showing physical world, art reveals inner world. Then it is not wrong to say that they have strong relations since they both have great effects on human beings. In this study, the relation between art and mathematics are shown by the examples of, Golden Ratio, Fibonacci sequence, and music. Talking about the relation between mathematics and arts, we frequently coincide some mathematical facts such as Golden Ratio and Fibonacci sequence. In fact, we see their applications on many items and even in nature. In this study, I will try to investigate the relation between mathematics and art and these concepts in detail. Word Count: 289 1 ZEYNEP AYGAR D1129018 CONTENTS ABSTRACT ................................................................................................................. 1 1. INTRODUCTION.................................................................................................. 3 2. GOLDEN RATIO.................................................................................................. 3 2.1. Definition..................................................................................................... 3 2.2. Applications.................................................................................................. 6 2.2.1. 3 lines.................................................................................................... 6 2.2.2. 3 sides: Triangle................................................................................... 7 2.2.3. 4 sides: Square..................................................................................... 7 2.2.4. Pyramid ................................................................................................ 8 3. GOLDEN RATIO IN ARHITECTURE............................................................. 9 3.1. Pyramids of Egypt..................................................................................... 9 3.2. Parthenon in Athens................................................................................. 10 3.3. Modern Buildings...................................................................................... 102 3.4. Islamic World............................................................................................ 13 4. FIBONACCI SEQUENCE.................................................................................. 14 5. MATHEMATICS IN NATURE.......................................................................... 15 6. GOLDEN RATIO IN PAINTING....................................................................... 18 7. MATHEMATICS IN MUSIC.............................................................................. 22 8. CONCLUSION...................................................................................................... 25 9. REFERENCES....................................................................................................... 26 2 ZEYNEP AYGAR D1129018 RESEARCH QUESTION Are mathematics and art related? 1. INTRODUCTION For some people, mathematics and arts are two different field of study which have no relations at all. In reality, this is not so. Mathematics has aesthetic, and art in itself and it has interrelation with the different fields or arts such as architecture, painting, poetry and music, and even in nature. While investigating the relation between mathematics and art, we frequently face with some mathematical concepts such as Golden Ratio, Fibonacci sequence, and Pythagoras Comma etc. How do these concepts are related to arts? Before going further, I will briefly describe what these concepts are. 2. GOLDEN RATIO 2.1. Definition The golden ratio, indicated by φ, is a geometric and numeric ratio of the parts of a whole that gives the most appropriate dimensions. It can be also called golden mean, divine proportion or golden section. Two objects are said to be in golden ratio if the sum of two objects to the larger one equals the ratio of the larger one to the smaller one. This can be simply explained on a line as explained in Figure 1 below: Figure 1: Golden Ratio on a line. 3 ZEYNEP AYGAR D1129018 This explanation of golden ratio was first given by Euclid ca. 300 BC in an equivalent definition so-called "extreme and mean ratios". Euclid stated that the total length of a line divided by the longer part is equal to longer part divided by shorter part. Thus, a perfect division is generated. This ratio is written as a2 – ab – b2 = 0 (1) If we take b = 1 then we have a2 – a – 1 = 0 (2) The solution of this quadratic equation gives that a = (1 + √5)/2 (3) = 1.61803398874989484820458634365638117720 ... (4) This number is called golden ratio and is indicated by the Greek letter φ. The reciprocal of golden ratio φ is called “phi” and for larger quantities of golden ratio is defined by “Phi”. Phi has connections with continued fractions and Euclidian algorithm for computing the greatest common divisor of two integers. When we use the same logic on a two dimensional objects such as a rectangle with dimensions of φ and 1, we get the rectangle shown on Figure 2(a). (a) (b) Figure 2 : Golden rectangle. Now, if we draw a square with length 1 within the rectangle of Figure 2(a), we get a square of size 1 and a new rectangle within the original one as shown in Figure 2(b) above. Notice that 4 ZEYNEP AYGAR D1129018 the yellow colored rectangles of two sketching Figure 2(a) and 2(b) are similar. Then if we apply Euclid’s rule in Figure 2(b), we get φ/1 = 1/(φ-1) (5) Then we have φ2 -φ - 1 = 0 (6) Notice that equations (2) and (6) are the same. So for the quadratic equations in (6) we get the same solution as in equation (3) resulting φ = (1 + √5)/2 (7) = 1.61803398874989484820458634365638117720 ... (8) The rectangle with the properties given above is called Golden Rectangle, which gives the most aesthetic view for the eye. In order to obtain a golden rectangle, a square with side 1 is drawn and the base line is extended outside of the square. From the middle point of the base of the square, a semicircle with the radius √5/2 is drawn by use of a compass. Notice that this semicircle passes over the upper corners of the square. If we complete the rectangle from the point that semicircle cuts the extended base line, we obtain a perfect golden rectangle as shown in the Figure 3 below. Figure 3 : Sketching golden rectangle. As seen from the Figure3, if one side (shorter side) of the rectangle is 1 unit long, then the longer side is (1 + √5)/2 in length, as calculated from the quadratic equations given in the equations (2) and (6). 5 ZEYNEP AYGAR D1129018 Successive points dividing a golden rectangle into squares and smaller rectangle as drawn on Figure 2(b) gives a logarithmic spiral that is mostly called the golden spiral, as shown on Figure 4 below. Figure 4 : Golden Spiral. 2.2. Applications The great German mathematician and astronomer Johannes Kepler described the golden number as one of the “two great treasures of geometry.” (The other is the Theorem of Pythagoras.) Golden number appears in many basic geometric constructions and applications: 2.2.1. 3 lines On a two dimensional space, if we take 3 equal lines and put one on Y axis, then put the 2nd line against the midpoint of the 1st line to X axis, and once more put the 3rd line against the midpoint of the 2nd. Line to X axis, we obtain the construction shown on Figure 5. In this figure, the ratio of AG to AB is Golden Number. 6 ZEYNEP AYGAR D1129018 Figure 5 : Golden Ratio obtained by 3 equal lines. 2.2.2. 3 sides: Triangle If we insert an equilateral triangle inside a circle and add a line at the midpoint of the two sides of the triangle and extend this line to the circle surrounding the triangle as in Figure 6, the ratio of line segment AG to line segment AB is Golden Number. Figure 6 : Golden Ratio in a triangle within a circle. 2.2.3. 4 sides: Square In a similar way, if we insert a square inside a semi-circle as in Figure 7, we find that the ratio of line segments AG to AB is Golden Number. Figure 7 : Golden Ratio in a square within a circle. 7 ZEYNEP AYGAR D1129018 2.2.4. Pyramid In a square pyramid